a We want to solve the given equation for x. To do this, we can start by using factoring. Then, we will use the Zero Product Property.
Factor the Expression
To factor a trinomial with a leading coefficient of 1, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.
x^2+6x- 40=0
In this case, we have - 40. This is a negative number, so for the product of the constant terms in the factors to be negative, these constants must have the opposite sign (one positive and one negative.)
Factor Constants
Product of Constants
- 1 and 40
- 40
1 and -40
- 40
- 2 and 20
- 40
2 and -20
- 40
- 4 and 10
- 40
4 and -10
- 40
- 5 and 8
- 40
5 and -8
- 40
Next, let's consider the coefficient of the linear term.
x^2+6x- 40=0
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, 6.
Factors
Sum of Factors
- 1 and 40
39
1 and -40
- 39
- 2 and 20
18
2 and -20
- 18
- 4 and 10
6
4 and -10
- 6
- 5 and 8
3
5 and -8
- 3
We found the factors whose product is - 40 and whose sum is 6.
x^2+6x- 40=0
⇕
(x-4)(x+10)=0
Zero Product Property
Since the equation is already written in factored form, we can now use the Zero Product Property.
Again, we created a true statement. x=- 10 is indeed a solution of the equation.
b We want to solve the given equation for x. To do this, we can start by using factoring. Then, we will use the Zero Product Property.
Factor the Expression
Here we have a quadratic trinomial of the form ax^2+bx+c, where |a| ≠1 and there are no common factors. To factor this expression, we will rewrite the middle term, bx, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b.
2x^2+13x-24=0
⇕
2x^2+13x+(- 24)=0
We have that a= 2, b=13, and c=- 24. There are now three steps we need to follow in order to rewrite the above expression.
Find a c. Since we have that a= 2 and c=- 24, the value of a c is 2* (- 24)=- 48.
Find factors of a c. Since ac=- 48, which is negative, we need factors of a c to have opposite signs — one positive and one negative. Since their sum b=13, which is positive, the absolute value of the positive factor will need to be greater than the absolute value of the negative factor.
